If sec A − tan A = x for an acute angle A, use trigonometric identities to find an equivalent expression for x in terms of sec A and tan A.

Introduction / Context:
This trigonometry question checks your understanding of identities involving secant and tangent. You are given that sec A − tan A equals x for an acute angle A and asked to find another expression that represents the same quantity. Recognising and applying identities like sec^2 A − tan^2 A = 1 and related manipulations is essential for simplifying trigonometric expressions in competitive exams.

Given Data / Assumptions:

• Angle A is acute, so sec A and tan A are well defined and positive.
• sec A − tan A = x.
• We must express x in an equivalent form using sec A and tan A.
• Standard trigonometric identities are valid: sec^2 A − tan^2 A = 1 and sec^2 A = 1 + tan^2 A.

Concept / Approach:
A key identity is (sec A − tan A)(sec A + tan A) = sec^2 A − tan^2 A. Since sec^2 A − tan^2 A equals 1, the product of sec A − tan A and sec A + tan A is 1. This means that each factor is the reciprocal of the other. Using this identity, we can express x as the reciprocal of sec A + tan A, which gives a compact alternative expression for x.

Step-by-Step Solution:
Step 1: Start with the identity sec^2 A − tan^2 A = 1.Step 2: Factor the left side as a difference of squares: sec^2 A − tan^2 A = (sec A − tan A)(sec A + tan A).Step 3: Therefore (sec A − tan A)(sec A + tan A) = 1.Step 4: From the problem statement, sec A − tan A = x.Step 5: Substitute x into the identity: x × (sec A + tan A) = 1.Step 6: Solve for x by dividing both sides by (sec A + tan A): x = 1 / (sec A + tan A).

Verification / Alternative check:
As a simple numerical example, take A = 60°. Then sec 60° = 2 and tan 60° = √3. Compute sec A − tan A = 2 − √3 and sec A + tan A = 2 + √3. The product (2 − √3)(2 + √3) equals 4 − 3 = 1, confirming the identity. Therefore sec A − tan A is indeed the reciprocal of sec A + tan A. This verifies that x = 1/(sec A + tan A) is correct.

Why Other Options Are Wrong:
The option 1/(sec^2 A − tan^2 A) simplifies to 1/1 = 1, which is not equal to sec A − tan A in general. The option 1/(sec^2 A + tan^2 A) does not simplify in a useful way and is not equivalent to x. The expression 1/√(sec^2 A − tan^2 A) becomes 1/1 = 1 again and fails for the same reason. The product sec A tan A has a completely different form and is not reciprocally related to sec A − tan A. Only 1/(sec A + tan A) follows directly from the factorised identity and equals x.

Common Pitfalls:
Some learners misremember the identity and think sec^2 A + tan^2 A equals 1, which leads them to choose an incorrect denominator. Others try to square or square root expressions without a clear plan, creating more complicated forms instead of simplifying. Focusing on the standard identity sec^2 A − tan^2 A = 1 and recognising it as a difference of squares helps quickly arrive at the correct reciprocal relationship.

Final Answer:
The expression equivalent to x when sec A − tan A = x is 1 / (sec A + tan A).